Center for Mathematical Sciences

/ Hiroyuki Sagawa / Professor
/ Hisasi Morikawa / Professor
/ Ken-ichi Funahashi / Associate Professor
/ Katsutaro Shimizu / Associate Professor
/ Kazuto Asai / Assistant Professor
/ Michio Honma / Assistant Professor
/ Shigeru Watanabe / Assistant Professor
/ Toshiro Watanabe / Assistant Professor
/ Hiroshi kihara / Research Associate

The scope of activities of the Center for Mathematical sciences spans all aspects of education and research in the fields of mathematical sciences. Current research directions in the field of mathematics are joined by the common theme ``Geometrical Method in Mathematical Sciences". In the fields of Physics, theoretical research is performed in many-body theories, Nuclear Physics and Quantum Gravity. Together with this, there is a project to develop educational software on quantum physics. The research areas assigned to each co-researcher are as follows:

• Prof. H. Morikawa investigates the theory of geometrical transformations, the algebraic methodology.

• Prof. H. Sagawa studies the physics of the many-body system of including nuclei and microclusters.

• Prof. K. Funahashi researches the theory of the neural networks and function theory of several complex variables from a geometrical viewpoint.

• Prof. K. Shimizu advances the traditional quantum theory and creates the geometrical theory of quantum gravity.

• Prof. T. Watanabe generalizes the unimodality problems of 1-dimensional infinitely decomposable distributions to multi-dimensional cases in the use of geometrical methods as the theory of manifolds.

• Prof. K. Asai researches combinatorial identities for geometrically generalized tableaux, and the connection between graphical compositions of indeterminate functions of several variables and homogeneous P.D.E. with constant coefficients.

• Prof. S. Watanabe studies geometrical interpretations of generating functions for spherical functions on homogeneous spaces.

• Prof. M. Honma researches the microscopic structures and dynamics of the nuclei by the algebraic methods and geometrical models performing the quantitative analysis by the large-scale numeric calculations.

• Prof. H. Kihara studies higher dimensional differential topology and elucidates the role of higher dimensional phenomena in various areas of mathematical sciences.

• Prof. S. Duzhin unifies algebraic topology, differential geometry and combinatorics with his investigations of nonlinear differential equations, singularity theory and knot invariants.

• Prof. A. Belyaev studies homogenization theory for partial differential equations in media with periodic structures. His research includes also applications of methods of differential geometry and partial differential equations in shape and image analysis.

Refereed Journal Papers

1. H. Katagawa, N. Tajima and H. Sagawa, Reaction Cross Sections and Radii of A=17 and A=20 isobar. Zeitschrift f\"{u}r Physik, vol.A358, p.381-387, 1997.

   Reaction cross sections and root-mean-square (rms) radii of $A$=17 and $A$=20 isobars are calculated by using a simplified Glauber model and spherical and deformed Hartree-Fock (HF) wave functions. The small separation energy effect is discussed on the reaction cross sections of $A$=17 isobars and possible proton halo in $^{17}$Ne. The calculated rms mass radii of $A$=20 isobars are increased appreciably by the deformation effect and show a similar irregular pattern as a function of the isospin to that of the observed radii.

2. I. Hamamoto, H. Sagawa and X. Z. Zhang, Structure of Giant Quadrupole Resonances in Neutron Drip Line Nuclei. Phys. Rev., vol.C55, p.2361-2368, 1997.

   We study the giant quadrupole resonances in the neutron drip line nucleus, $^{28}_{8}O_{20}$~, using the self-consistent Hartree-Fock calculation plus the random phase approximation with Skyrme interactions. Including simultaneously both the isoscalar and the isovector correlation the RPA response function is calculated in the coordinate space so as to take into account properly the continuum effect. Though in neutron drip line nuclei the distribution of the quadrupole strength is much affected by the presence of the low-energy threshold strength, the isoscalar and the isovector correlation structure as well as the transition densities of the isoscalar giant resonance are in good agreement with those given by the collective model. In contrast, the transition density of the isovector giant resonance shows a clear deviation from that of the liquid-drop polarization mode.

3. N. Yoshida, A. Gelberg, T. Otsuka, I. Wiedenhover, H. Sagawa and P. von Brentano, IBFM2 Description of odd Xe and Cs isotopes. Nucl. Phys., vol.A691, p.65-87, 1997.

   Excitation energies, branching ratios and E2/M1 mixing ratios of the negative-parity states in $^{125,127}$Xe and $^{125,127,129}$Cs have been calculated by using the proton-neutron Interacting Boson Fermion Model (IBFM2). A good agreement with the experiment has been obtained. The sensitivity of the calculated observables to changes in model parameters is discussed and the results are compared with those obtained from the IBFM1. We examine a symmetry of the hamiltonian with respect to the change of sign of the parameters $\chi_{\pi},\chi_{\nu}$ and $\Gamma$. The different behaviours of the odd Xe and Cs isotopes arise from the different position of the proton and neutron Fermi levels. In particular the exchange term of the hamiltonian in $^{125,127,129}$Cs can be neglected.

4. A. Odahara, Y. Gono, H. Sagawa et al., High-spin Isomers and Level Structure of $^{145}$Sm. Nucl. Phys., vol.A620, p.363-384, 1997.

   High-spin states of $^{145}$Sm were studied by using several heavy ion reactions. The excitatione energy and spin-parity of high spin isomers are determined by using the deformed independent particle model.

5. I. Hamamoto, H. Sagawa and X. Z. Zhang, Quadrupole Strength Function and Core Polarization in Drip Line Nuclei. Nucl. Phys., vol.A626, p.669-685, 1997.

   Using the self-consistent Hartree-Fock calculation plus RPA with Skyrme interactions, the RPA quadrupole strength function is estimated in the coordinate space, including simultaneously both the isoscalar and the isovector correlation. We discuss the result of the isoscalar, the isovector and the electric quadrupole polarization of the Ca-isotopes from the proton drip line towards the neutron drip line. We study also the comparison of the polarizations in the $A\!=\!48$ mirror nuclei, $^{48}_{28}Ni_{20}$ and $^{48}_{20}Ca_{28}$~, and the dependence of the polarizations of $^{28}_{8}O_{20}$ on various Skyrme interactions.

6. I. Hamamoto, H. Sagawa and X. Z. Zhang, Giant Monopole States in Static and Dripline Nuclei. Phys. Rev., vol.C56, p.3121-3133, 1997.

   Isoscalar and isovector monopole responses of Ca-isotopes towards drip lines are studied in comparison with those of stable nuclei, $^{40}$Ca, $^{90}$Zr and $^{208}$Pb~, using the self-consistent Hartree-Fock calculation plus the random phase approximation with Skyrme interactions. Including simultaneously both the isoscalar and the isovector correlation the RPA response function is calculated in the coordinate space so as to take properly into account the continuum effect. The distribution of the monopole strength is much affected by the presence of the low-energy threshold strength in both proton and neutron drip line nuclei, while the isoscalar monopole strength is concentrated in one giant peak in the heavy $\beta$-stable nucleus $^{208}$Pb. It is found that in drip line nuclei the transition density and the displacement field of protons are very different from those of neutrons, especially around the nuclear surface. The relation between the compression modulus of the nuclear matter $K_{nm}$ and that of finite nuclei $K_{A}$ is also discussed by using the energy moments estimated in RPA.

7. Toshio Suzuki, H. Sagawa and Nguyen Van Giai, Electric Dipole Transitions between Gamow-Teller and Spin-Dipole States. Phys. Rev., vol.C57, p.139-144, 1998.

   We study electric dipole transitions between Gamow-Teller (GT) and spin-dipole (SD) states. SD and GT excitations are calculated within the Hartree-Fock (HF) + Tamm- Dancoff approximation (TDA) for $^{48}$Sc and $^{90}$Nb. The electric dipole transitions are found to be rather selective and strong E1 transitions occur to some specific spin-dipole states. Calculated E1 transition strengths between GT and SD states are compared with the analytic sum rules within 1 particle-1 hole (1p-1h) configuration space and within both 1p-1h and 2p-2h model space. Possible implications for charge-exchange reactions may help to understand the quenching problem of spin excitations.

8. C. A. Bertulani, P. Lotti, and H. Sagawa, Energy Dependence of Breakup Cross Sections of Halo Nucleus $^8B$ and Effective Interactions. Phys. Rev., vol.C57, p.217-222, 1998.

   We study the energy dependence of the cross sections for nucleon removal of $^8B$ projectiles. It is shown that the Glauber model calculations with nucleon-nucleon t-matrix reproduce well the energy dependence of the breakup cross sections of $^8B$. A DWBA model for the breakup cross section is also proposed and results are compared with those of the Glauber model. We show that to obtain an agreement between the DWBA calculations, the Glauber formalism, and the experimental data, it is necessary to modify the energy behavior of the effective interaction. In particular, the breakup potential has a quite different energy dependence than the strong absorption potential.

9. I. Hamamoto, H. Sagawa and X. Z. Zhang, Isoscalar and Isovector Dipole Mode in Drip Line Nuclei. Phys. Rev., vol.C57, p.R1064-1068, 1998.

   The isoscalar and the isovector dipole mode of drip line and $\beta$-stable nuclei are investigated, using the self-consistent Hartree-Fock plus the random-phase approximation with Skyrme interactions. Including simultaneously both the isoscalar and the isovector correlation, the RPA response function is estimated in the coordinate space so as to take into account properly the continuum effect. The spurious component is carefully taken away from the calculated strength. In $\beta$-stable nuclei such as ~$^{208}Pb$~ the frequency of the isovector giant dipole resonance (IVGDR) is lower than that of the isoscalar giant dipole resonance (compression mode). In contrast, in lighter drip line nuclei a major part of the isoscalar compression dipole strength lies at an energy much lower than the energy of the IVGDR.

10. Funahashi, K., Multilayer neural networks and Bayes decision theory. Neural Networks, vol.11, no.2, p.209--213, 1998.

11. Watanabe, T., Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type. J. Math. Soc. Japan, vol.50, no.1, p.155--168, 1998.

12. Shigeru Watanabe, Generating functions of the Jacobi polynomials and related Hilbert spaces of analytic functions. Proceedings of the Japan Academy, 1998, accepted.

Refereed Proceeding Papers

1. H. Sagawa, Giant Monopole States in Nuclei near Drip Lines. Proc. of the Third INFN-RIKEN Meeting ``Perspectives in Heavy Ion Physics", 1997.

2. I. Hamamoto, H. Sagawa and X. Z. Zhang, Structure of Nuclei near Drip Lines. Proc. of Int. Workshop on ``Unstable Nuclei", 1998.

3. H. Sagawa, I. Hamamoto and X. Z. Zhang, Giant Monopole States in Nuclei near Drip Lines. Proc. of Int. Workshop on ``Unstable Nuclei", 1998.

4. H. Sagawa, Soft and Giant Multipole States in Nuclei near Drip Lines. Proc of Erice School on ``Heavy Ion Physics", p.151-170, 1997.

1. H. Sagawa and M. Honma, Electromagnetism, Springer-Verlag Tokyo, Tokyo, 1997.

2. H. Sagawa and K. Shimizu, Quantum Mechanics, Springer-Verlag Tokyo, Tokyo, 1998.

3. Hiroyuki Sagawa and Katsutaro Shimizu, Quantum Mechanics, Springer-Verlag Tokyo, Tokyo, 1998.

1. Hiroyuki Sagawa, Ministry of Education Scientific Research Fund. bases(c), Physics, No.09640369. 1997.

Academic Activities

1. Toshiro Watanabe, A member of working committee of Math. Soc. Japan (1998.3 - ).

2. Hiroyuki Sagawa, referee of Physical Review C.

3. Hiroyuki Sagawa, referee of Physical Review Letters.

4. Hiroyuki Sagawa, referee of Prog. of Theoretical Physics.

Others

1. Takashi Atzumi, Quantum Cryptography, The Univ. of Aizu, 1997. Thesis Advisor: Hiroyuki Sagawa.

2. T. Naito, Factorization of Number by Quantum Computer, The Univ. of Aizu, 1997. Thesis Advisor: Hiroyuki Sagawa.

3. H. Yagi, Prediction of Economics as a Complex System, The Univ. of Aizu, 1997. Thesis Advisor: Hiroyuki Sagawa.

4. S. Nakaune, Quantum Logic Gate, The Univ. of Aizu, 1997. Thesis Advisor: Hiroyuki Sagawa.

5. Shinya Yamabe, Fractal Structure in Economy, The Univ. of Aizu, 1997. Thesis Advisor: Hiroyuki Sagawa.